Discrete dynamics and differentiable stacks

TL;DR

This paper explores how the orbit stacks of discrete group actions on manifolds encode the dynamics, generalizing previous results and applying stack theory to problems in geometric dynamics and topology.

Contribution

It establishes that orbit stacks encode discrete dynamics up to conjugation and inversion, extending to non-simply connected manifolds and connecting to stack covering theory.

Findings

Orbit stacks encode dynamics up to conjugation and inversion.

Generalization of dynamics encoding to arbitrary discrete groups and non-simply connected manifolds.

Applications include a geometric version of Rieffel's theorem, computation of stack fundamental groups, and classification of lens spaces.

Abstract

In this paper we relate the study of actions of discrete groups over connected manifolds to that of their orbit spaces seen as differentiable stacks. We show that the orbit stack of a discrete dynamical system on a simply connected manifold encodes the dynamics up to conjugation and inversion. We also prove a generalization of this result for arbitrary discrete groups and non-simply connected manifolds, and relate it to the covering theory of stacks. As applications, we obtain a geometric version of Rieffel's theorem on irrational rotations of the circle, we compute the stack-theoretic fundamental group of hyperbolic toral automorphisms, and we revisit the classification of lens spaces.